Continuous Functions Basics

Continuous functions in topology are maps between spaces that preserve their structure, allowing deformations without breaks. This guide from MathMultiverse explores the definition, properties, and examples of continuous functions, providing a clear foundation for understanding this fundamental concept in topology.

Definition

A function \(f: X \to Y\) between topological spaces \(X\) and \(Y\) is continuous if for every open set \(V \subseteq Y\), the preimage \(f^{-1}(V)\) is open in \(X\). This ensures that the function respects the topological structure of the spaces.

\[ f: X \to Y \text{ is continuous} \iff \forall V \subseteq Y \text{ open}, f^{-1}(V) \subseteq X \text{ is open} \]

Examples

The function \(f(x) = x^2\) from \(\mathbb{R}\) to \(\mathbb{R}\) (with the standard topology) is continuous, as the preimage of any open interval \((a, b)\) in \(\mathbb{R}\) is an open set in \(\mathbb{R}\). For instance, \(f^{-1}((a, b)) = (-\sqrt{b}, -\sqrt{a}) \cup (\sqrt{a}, \sqrt{b})\) for \(a, b > 0\), which is open.

\[ f(x) = x^2, \quad f^{-1}((a, b)) = (-\sqrt{b}, -\sqrt{a}) \cup (\sqrt{a}, \sqrt{b}) \text{ for } a, b > 0 \]

Properties

Continuous functions preserve key topological properties. If \(X\) is connected, then \(f(X)\) is connected. If \(X\) is compact, then \(f(X)\) is compact. Additionally, continuous functions map compact sets to bounded sets that attain their bounds in metric spaces.

\[ f: X \to Y \text{ continuous}, X \text{ compact} \implies f(X) \text{ is compact} \]